Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 343

Substituting

vk=rkexp(iαk)

into (8.89) we infer that the real functionsrkandαkobey the condition

rk^2 α′k= 1. (8.90)

Next we note that (8.87) describes a harmonic oscillator with energy

Ek=

1

2

(∣

∣vk′

∣

∣^2 +ωk^2 |vk|^2

)

=

1

2

(

rk′^2 +rk^2 αk′^2 +ω^2 krk^2

)

=

1

2

(

rk′^2 +

1

r^2 k

+ω^2 krk^2

)

. (8.91)

We want to consider the minimal possible fluctuations allowed by the uncertainty
relations. The energy is minimized whenrk′(ηi)=0 andrk(ηi)=ω−k^1 /^2. We thus
obtain

vk(ηi)=

1

√

ωk

eiαk(ηi),v′k(ηi)=i

√

ωkeiαk(ηi). (8.92)

Although the phase factorsαk(ηi)remain undetermined, they are irrelevant and we
can set them to zero. Note that the above considerations are valid only ifω^2 k>0,
that is, for modes withc^2 sk^2 >

(

z′′/z

)

i.
The next step in quantization is to define the “vacuum” state| 0 〉as the state
annihilated by operatorsaˆ−k:

aˆk−| 0 〉= 0. (8.93)

We further assume that a complete set of independent states in the corresponding
Hilbert space can be obtained by acting with the products of creation operators on the
vacuum state| 0 〉.Iftheωkdo not depend on time, then the vector| 0 〉corresponds
to the familiar Minkowski vacuum.Assumingcschanges adiabatically, we find
that modes withc^2 sk^2 

(

z′′/z

)

remain unexcited and minimal fluctuations are well
defined. On the other hand, for modes withc^2 sk^2 <

(

z′′/z

)

iwe haveω

2
k(ηi)<0, and
the initial minimal fluctuations on corresponding scales cannot be unambiguously
determined. These scales exceed the Hubble scale at the beginning of inflation and
are subsequently stretched to huge unobservable scales; therefore the question of
initial fluctuations here is fortunately moot. The inhomogeneities responsible for
the observable structure originate from quantum fluctuations on scales where the
minimal fluctuations are unambiguously defined.

SpectrumOur final task is to calculate the correlation function, or equivalently, the
power spectrum of the gravitational potential. Taking into account (8.56), we have